1 Since the estimates in this publication are based on information obtained from occupants of a sample of dwellings, they are subject to sampling variability. That is, they may differ from those estimates that would have been produced if all dwellings had been included in the survey. One measure of the likely difference is given by the standard error (SE), which indicates the extent to which an estimate might have varied by chance because only a sample of dwellings was included. There are about two chances in three (67%) that a sample estimate will differ by less than one SE from the number that would have been obtained if all dwellings had been included, and about 19 chances in 20 (95%) that the difference will be less than two SEs. Another measure of the likely difference is the relative standard error (RSE), which is obtained by expressing the SE as a percentage of the estimate.

2 Due to space limitations, it is impractical to print the SE of each estimate in the publication. Instead, a table of SEs is provided to enable readers to determine the SE for an estimate from the size of that estimate (see table T1). The SE table is derived from a mathematical model, referred to as the 'SE model', which is created using data from a number of past Labour Force Surveys. It should be noted that the SE model only gives an approximate value for the SE for any particular estimate, since there is some minor variation between SEs for different estimates of the same size.

CALCULATION OF STANDARD ERROR

3 An example of the calculation and the use of SEs in relation to estimates of persons is as follows. Table 4 shows the estimated number of unemployed females in Australia who were looking for full-time work was 165,200. Since this estimate is between 150,000 and 200,000, table T1 shows that the SE for Australia will lie between 5,000 and 5,600 and can be approximated by interpolation using the following general formula:

4 Therefore, there are about two chances in three that the value that would have been produced if all dwellings had been included in the survey will fall within the range 160,000 to 170,400 and about 19 chances in 20 that the value will fall within the range 154,800 to 175,600. This example is illustrated in the diagram below.

5 In general, the size of the SE increases as the size of the estimate increases. Conversely, the RSE decreases as the size of the estimate increases. Very small estimates are thus subject to such high RSEs that their value for most practical purposes is unreliable. In the tables in this publication, only estimates with RSEs of 25% or less are considered reliable for most purposes. Estimates with RSEs greater than 25% but less than or equal to 50% are preceded by an asterisk (e.g. *3.4) to indicate they are subject to high SEs and should be used with caution. Estimates with RSEs of greater than 50%, preceded by a double asterisk (e.g. **0.2), are considered too unreliable for general use and should only be used to aggregate with other estimates to provide derived estimates with RSEs of less than 25%. Table T2 presents the levels at which estimates have RSEs of 25% and 50%.

MEANS AND MEDIANS

6 The RSEs of estimates of mean duration of unemployment and median duration of unemployment are obtained by first finding the RSE of the estimate of the total number of persons contributing to the mean or median (see table T1) and then multiplying the resulting number by the following factors for Australian estimates:

mean duration of unemployment: 1.6

median duration of unemployment: 2.5

7 The following is an example of the calculation of SEs where the use of a factor is required. Table 4 shows that the estimated median duration of unemployment for unemployed females in Australia was 17 weeks and shows that the number of unemployed females was estimated as 271,900. The SE of 271,900 can be calculated from table T1 (by interpolation) as 6,400. To convert this to an RSE we express the SE as a percentage of the estimate or 6,400/271,900 =2.4%.

8 The RSE of the estimate of median duration of unemployment for unemployed females is calculated by multiplying this number (2.4%) by the appropriate factor shown in the previous paragraph (in this case 2.5): 2.4 x 2.5 = 6%. The SE of this estimate of median duration of unemployment for unemployed females is therefore 6% of 17 weeks, i.e. approximately one week. Therefore, there are two chances in three that the median duration of unemployment for females that would have been obtained if all dwellings had been included in the survey would have been within the range 16 to 18 weeks and about 19 chances in 20 that it would have been within the range 15 weeks to 19 weeks.

9 Table T2 represents the minimum size of estimates, based on the SE model described in paragraph 2, required to have RSEs of less than 25% and 50% respectively. For example, an estimate of median duration of unemployment for Australia based on less than 29,000 persons will have an RSE of at least 25%, and an estimate of median duration of unemployment for Australia based on less than 10,000 will have an RSE of at least 50%. For all other estimates, (i.e. those estimates based purely on number of persons in a specific category), an estimate of less than 6,800 for the Australian total will have an RSE of at least 25% and an estimate of less than 1,600 will have an RSE of at least 50%.

PROPORTIONS AND PERCENTAGES

10 Proportions and percentages formed from the ratio of two estimates are also subject to sampling errors. The size of the error depends on the accuracy of both the numerator and the denominator. A formula to approximate the RSE of a proportion is given below. This formula is only valid when x is a subset of y:

11 Considering the example from the previous page, of the 165,200 unemployed females who were looking for full-time work, 36,200 or 21.9% had been unemployed for one year or more. The SE of 36,200 may be calculated by interpolation as 3,100. To convert this to an RSE we express the SE as a percentage of the estimate, or 3,100/36,200 = 8.6%. The SE for 165,200 was calculated previously as 5,200, which converted to an RSE is 5,200/165,200 = 3.1%. Applying the above formula, the RSE of the proportion is:

12 Therefore, the SE for the proportion of unemployed females looking for full-time work who had been unemployed for one year or more is 1.8 percentage points (=(21.9/100)x8.0). Therefore, there are about two chances in three that the proportion of unemployed females looking for full-time work who have been unemployed for one year or more is between 20.1% and 23.7% and 19 chances in 20 that the proportion is within the range 18.3% to 25.5%.

DIFFERENCES

13 Published estimates may also be used to calculate the difference between two survey estimates (of numbers or percentages). Such an estimate is subject to sampling error. The sampling error of the difference between two estimates depends on their SEs and the relationship (correlation) between them. An approximate SE of the difference between two estimates (x-y) may be calculated by the following formula:

14 While this formula will only be exact for differences between separate and uncorrelated characteristics or subpopulations, it is expected to provide a good approximation for all differences likely to be of interest in this publication.

STANDARD ERRORS

T1 STANDARD ERRORS OF ESTIMATES

NSW

Vic.

Qld.

SA

WA

Tas.

NT

ACT

Aust.

Size of estimate (persons)

No.

No.

No.

No.

No.

No.

No.

No.

No.

%

100

290

290

220

180

220

110

80

100

110

110.0

200

400

380

320

240

290

160

120

170

190

95.0

300

470

440

390

280

340

190

150

210

260

86.7

500

580

540

500

340

420

240

200

270

380

76.0

700

660

620

580

390

480

270

230

300

480

68.6

1,000

760

710

680

450

550

310

270

330

610

61.0

1,500

900

830

810

530

640

360

320

360

780

52.0

2,000

1 010

930

910

590

710

390

350

390

920

46.0

2,500

1 100

1 000

1 000

650

800

400

400

400

1 050

42.0

3,000

1 200

1 100

1 050

700

850

450

400

450

1 150

38.3

3,500

1 250

1 150

1 100

700

900

450

400

450

1 250

35.7

4,000

1 300

1 200

1 200

750

900

500

450

450

1 350

33.8

5,000

1 450

1 300

1 250

800

1 000

500

500

500

1 500

30.0

7,000

1 650

1 500

1 450

900

1 150

600

600

600

1 700

24.3

10,000

1 850

1 700

1 600

1 050

1 300

700

750

700

2 000

20.0

15,000

2 150

1 950

1 800

1 200

1 500

850

1 000

850

2 350

15.7

20,000

2 400

2 200

1 950

1 350

1 650

1 000

1 300

1 000

2 550

12.8

30,000

2 800

2 550

2 250

1 550

1 900

1 250

1 800

1 250

2 900

9.7

40,000

3 100

2 800

2 500

1 800

2 100

1 500

2 300

1 500

3 150

7.9

50,000

3 350

3 050

2 750

2 000

2 300

1 700

2 750

1 650

3 400

6.8

100,000

4 250

4 000

3 750

3 000

3 400

2 400

4 750

2 250

4 300

4.3

150,000

5 000

4 850

4 600

3 850

4 450

2 850

6 500

2 500

5 000

3.3

200,000

5 750

5 650

5 400

4 550

5 350

3 200

8 150

2 650

5 600

2.8

300,000

7 250

7 250

6 850

5 550

6 750

3 700

11 100

2 800

6 650

2.2

500,000

10 150

10 050

9 250

7 000

8 600

4 250

. .

2 800

8 350

1.7

1,000,000

15 100

15 250

13 200

8 900

10 950

4 850

. .

. .

11 750

1.2

2,000,000

20 350

22 550

17 700

10 600

12 700

. .

. .

. .

17 050

0.9

5,000,000

25 900

36 100

23 900

11 900

13 250

. .

. .

. .

28 450

0.6

10,000,000

27 750

49 750

27 950

. .

. .

. .

. .

. .

37 950

0.4

15,000,000

. .

. .

. .

. .

. .

. .

. .

. .

42 850

0.3

. . not applicable

T2 Levels at which estimates have relative standard errors of 25% and 50%(a)